LiMM-PCA with the Human Serum Data

1 Data set up

# serum human data base 5
load(file.path(data_path, "HumanSerumData.RData"))

n <- nrow(design)

ppm <- colnames(outcomes)

1.1 Article graph

# pdf(file.path(fig_path, "HSD_outcomes.pdf"), width = 8, height = 5)
col <- gg_color_hue(dim(outcomes)[1])
plot(as.numeric(ppm), outcomes[1,], 
     col=col[1], ylim=range(outcomes),ylab="Intensity",
     main ="Human serum data base spectra", type="l",
      xlab="ppm", xlim=rev(range(as.numeric(ppm))))
for (i in 2:dim(outcomes)[1]){
  lines(as.numeric(ppm), outcomes[i,], col=col[i])
}

# dev.off()

1.2 Remove volunteers?

if (RemVol) {
  outcomes <- outcomes[which(!design$Volunteer %in% 
                               c("05", "04")),]
  design <- design[which(!design$Volunteer%in% c("05", "04")),]
  design$Volunteer
  design$Volunteer <- droplevels(design$Volunteer)
}

1.3 Data dimensions and frequencies

pander("dim(design)")

dim(design)

pander(paste(dim(design),collapse = " x "))

140 x 6

pander("dim(outcomes)")

dim(outcomes)

pander(paste(dim(outcomes),collapse = " x "))

140 x 750

pander("table(design$Tube, design$Sampling, design$Time)")

table(design\(Tube, design\)Sampling, design$Time)

pander(table(design$Tube, design$Sampling, design$Time))

			1	2
1	1		12	12
	2		12	12
	3		12	12
2	1		11	11
	2		11	11
	3		12	12

spectmat <- outcomes

Volunteer  <- design$Volunteer
Sampling <- design$Sampling
Tube <- design$Tube
Time <- design$Time

1.4 Hotelling T^2

require(mdatools)

## Loading required package: mdatools

## Registered S3 methods overwritten by 'mdatools':
##   method         from
##   plot.randtest  ade4
##   print.randtest ade4

model = pca(spectmat, scale = FALSE, cv = 10, 
            info = 'Simple PCA model', lim.type = "jm")

ncomp <- 4
# model$ncomp.selected

# plotVariance(model)
# plotResiduals(model, show.labels = TRUE, ncomp = ncomp, main = "Squared residual distance vs Hotelling T2 distance", norm = TRUE)
# 
# plotResiduals(model, show.labels = TRUE, ncomp = ncomp, 
#               main = "Squared residual distance vs Hotelling T2 distance")


Qlim <- model$Qlim
T2lim <- model$T2lim
rownames(Qlim)[2] <- rownames(T2lim)[2] <- "Out_limit"


# In case of PCA the critical limits are just shown 
# on residual plot as lines and can be used for detection 
# of extreme objects (solid line) and outliers (dashed line).

plot_hotelling <- function(){
  xlim <- range(model$calres$T2[,ncomp])
  xlim[1] <- xlim[1]*0.9
  xlim[2] <- xlim[2]*1.1
  ylim <- range(model$calres$Q[,ncomp])
  ylim[1] <- ylim[1]*0.9
  ylim[2] <- ylim[2]*1.1
  plot(model$calres$T2[,ncomp], model$calres$Q[,ncomp], 
     main = "Diagnostic plot for score and residual outliers", xlab = "Hotelling T2 distance", 
     ylab ="Squared residual distance", pch = 16, xlim = xlim, ylim = ylim)
  abline(h=Qlim["Out_limit",ncomp], v=T2lim["Out_limit",ncomp], lty =3)
  legend("topright", legend = "Outlier limit", lty = 3)
  index1 <- which(model$calres$T2[,ncomp]>=T2lim["Out_limit",ncomp])
  index2 <- which(model$calres$Q[,ncomp]>=Qlim["Out_limit",ncomp])
  index_ho <- unique(c(index1,index2))
  text(x = model$calres$T2[index_ho,ncomp], y=model$calres$Q[index_ho,ncomp], 
       labels = names(model$calres$T2[index_ho,ncomp]), pos = c(1,2,3,4))
}

plot_hotelling()

# pdf(file.path(fig_path,"HSD_hotelling.pdf"), height = 6, width = 6)
plot_hotelling()

# dev.off()

2 Dimension reduction by PCA

###################################################
# Dimension reduction by PCA
###################################################

# ===== PCA ===== #

res_pca <- SVDforPCA(x = spectmat)

### screePlot
df <- data.frame(PC =as.character(1:length(res_pca$var)),
                 var = res_pca$var)
df <- df[c(1:9),]
screePlot <- ggplot(df, aes(y=0,yend=var,x=PC,
                          xend=PC))+ geom_segment()+
          labs(title= "PCA scree plot",
                        x = "PC", y="% var")+theme_classic()



### score plot
score1 <- DrawScores(res_pca, axes=c(1,2),drawNames = FALSE, 
           main  = "PCA scores plot", pch = design$Time,
           color= design$Volunteer)+
  coord_cartesian(xlim = c(-30, 55), ylim = c(-15, 17))



col <- gg_color_hue(2)
score2 <- DrawScores(res_pca, axes=c(1,2),drawNames = FALSE, 
           main  = "PCA scores plot", 
           pch = Volunteer,
            size = 3, 
           legend_color_manual = c("1"=col[1], "2"=col[2])) +
  coord_cartesian(xlim = c(-51, 51), ylim = c(-18, 18))



# pdf(file.path(fig_path,"ScoresSpectmat.pdf"), height = 3.2, width = 9.4)

# plot_grid(score1,score2, align = "none", nrow = 1)

# dev.off()

score1
score2

DrawScores(res_pca, color = design$Sampling, 
           axes=c(1,2), drawNames = TRUE)
DrawScores(res_pca, color = design$Sampling, 
           axes=c(3,4), drawNames = TRUE)
DrawScores(res_pca, color = design$Volunteer, 
           axes=c(1,2), drawNames = FALSE)
DrawScores(res_pca, color = design$Volunteer, 
           axes=c(3,4), drawNames = FALSE)
DrawScores(res_pca, color = design$Tube, axes=c(1,2), drawNames = FALSE)
DrawScores(res_pca, color = design$Tube, axes=c(3,4), drawNames = FALSE)
DrawScores(res_pca, color = design$Time, axes=c(1,2), drawNames = FALSE)
DrawScores(res_pca, color = design$Time, axes=c(3,4), drawNames = FALSE)


pander("pc var explained")

pc var explained

pander(round(res_pca$var[1:8],2))

PC1	PC2	PC3	PC4	PC5	PC6	PC7	PC8
77.12	8.95	5.55	2.23	1.33	0.97	0.64	0.52

2.1 Article journal graph

loadPlot <- DrawLoadings(res_pca, type.obj = "PCA", 
                         main = "PCA loadings plots",
             axes = c(1, 2), loadingstype = "s",ylab = c(""),
             xaxis_type="numeric", xlab = "ppm")[[1]]

PCA_plots <- ggarrange(screePlot, score2, loadPlot,
                ncol = 3, nrow = 1,
                widths  = c(0.3, 1,  0.85))

PCA_plots

# ggexport(PCA_plots, 
#          filename = file.path(fig_path,paste0(Vers,
#            "HSD_PCA_outcomes.pdf")),
#          height = 5, width = 14)

2.2 Apply first step of LiMMPCA (PCA)

# apply first step of LiMMPCA with dimreducPCA()
res_pca2 <- dimreducPCA(data = outcomes, pcvar = 99)


spectra_PCA_scores <- res_pca2$pca_scores
spectra_PCA_loadings <- res_pca2$pca_loadings


Xtest <- spectra_PCA_scores%*%t(spectra_PCA_loadings)

m <- nPC <- dim(spectra_PCA_scores)[2]

3 Parallel mixed modelling

###################################################
# Parallel mixed modelling
###################################################

### define the input arguments for the parallel mixed modelling

# formula
form <- "~ Time + Tube + (1|Volunteer) + (1|Volunteer:Sampling) "

# multivariate response matrix
outcomes <- spectra_PCA_scores

# REML
REML <- TRUE

### run parlmer
res.parlmer <- parlmer(design, outcomes, form, REML)
MM_full <- res.parlmer


# save the results
RanModMatlist <- res.parlmer$RanModMatlist
FixedModMatlist <- res.parlmer$FixedModMatlist

# fixef_names <- attr(terms(MM_full$merMod_obj[[1]], fixed.only=TRUE),"term.labels")

# randef_names <- attr(terms(MM_full$merMod_obj[[1]], random.only=TRUE),"term.labels")


# modmat_fixed <- model.matrix(MM_full$merMod_obj[[1]], type="fixed")
# modmat_random <- model.matrix(MM_full$merMod_obj[[1]], type="random")
# modmat_random <- as.matrix(modmat_random)
# rownames(modmat_random) <- rownames(modmat_fixed)

# Residuals sd error
Res_std_error_PC <- sapply(MM_full$merMod_obj, sigma)

# Residuals
Residuals_PC <- sapply(MM_full$merMod_obj, residuals)


### ranef_PC: Extract the modes of the random effects
ranef_PC <- sapply(MM_full$merMod_obj, function(x) unlist(ranef(x)))


# recover accurate rownames and colnames of ranef_PC
list_rownam <- lapply(ranef(MM_full$merMod_obj[[1]]), rownames)

colnam <- paste0(names(ranef(MM_full$merMod_obj[[1]])), lapply(ranef(MM_full$merMod_obj[[1]]), rownames))

ranef_PC <- sapply(MM_full$merMod_obj, function(x) unlist(ranef(x)))
names(list_rownam) <- gsub("[^A-z]", "", names(list_rownam))

colnam <- c()
for (i in 1:length(list_rownam)){
  colnam <- c(colnam, paste0(names(list_rownam)[i], list_rownam[[i]]))
}
rownames(ranef_PC) <- colnam
rownames(ranef_PC) <- gsub("..(Intercept))", "", rownames(ranef_PC))


### fixef: Extract fixed-effects estimates
fixef_PC <- sapply(MM_full$merMod_obj, fixef)


### all fixed estimates and random predictions
cof_PC <- rbind(fixef_PC, ranef_PC)

### Extract Variance and Correlation Components
varcor_random_full <- sapply(MM_full$merMod_obj, VarCorr) # var

dat <- as.numeric(rbind(varcor_random_full))
varcor_random_full_mat <- matrix(dat, nrow=dim(varcor_random_full)[1],
                                 dimnames = dimnames(varcor_random_full))


fixNames <- MM_full$fixNames # names of fixed effects

ranNames <- MM_full$ranNames # names of random effects

# estimated Std.Dev. of fixed effects parameters
varcor_fixed_full <- sapply(MM_full$merMod_obj, 
                            function(x) sqrt(diag(vcov(x)))) # Std.Dev.


rownames(varcor_fixed_full) <- rownames(vcov(MM_full$merMod_obj[[1]]))

4 Effect matrix computation

###################################################
# Effect matrix computation
###################################################

# fixed effects + intercept -----------------------------
dim1FixedModMad <- sapply(FixedModMatlist, function(x) dim(x)[2])
names_FixedEffects <- names(FixedModMatlist)

shortFixedNames <- names_FixedEffects
Xmat <- do.call(cbind, FixedModMatlist)

# colnames(Xmat) %in% rownames(fixef_PC)
Xmat <- Xmat[,rownames(fixef_PC)] # reorder colnames of Xmat

index <- cumsum(dim1FixedModMad)
k <- 1
Mfix <- vector("list", length=length(shortFixedNames))
Mfix_PC <- vector("list", length=length(shortFixedNames))
names(Mfix) <- names(Mfix_PC) <-  shortFixedNames
for (i in 1:length(shortFixedNames)){
  XMfix = Xmat
  XMfix[,-c(k:index[i])] = 0
  Mfix_PC[[i]] = XMfix %*% fixef_PC
  Mfix[[i]] <- Mfix_PC[[i]]%*%t(spectra_PCA_loadings)
  k <-  index[i] + 1
}

M0 <- Mfix$Intercept

Mfix <- Mfix[-which(names(Mfix)=="(Intercept)")]
Mfix_PC <- Mfix_PC[-which(names(Mfix_PC)=="(Intercept)")]


# random effects -----------------------------
dim1RandModMad <- sapply(RanModMatlist, function(x) dim(x)[2])
names_randomEffects <- names(RanModMatlist)

shortRandNames <- gsub("[^A-z]", "", names_randomEffects)

Zmat <- do.call(cbind, RanModMatlist)
colnames(Zmat) <- paste0(rep(shortRandNames,
                             dim1RandModMad),
                         colnames(Zmat))

# colnames(Zmat) %in% rownames(ranef_PC)

Zmat <- Zmat[,rownames(ranef_PC)] # reorder colnames of Zmat

index <- cumsum(dim1RandModMad)
k <- 1
Mrand_PC <- vector("list", length=length(ranNames))
Mrand <- vector("list", length=length(ranNames))
names(Mrand) <- names(Mrand_PC) <- ranNames
for (i in 1:length(shortRandNames)){
  XMrand = Zmat
  XMrand[,-c(k:index[i])] = 0
  Mrand_PC[[i]] = XMrand %*% ranef_PC
  Mrand[[i]] <- Mrand_PC[[i]]%*%t(spectra_PCA_loadings)
  k <-  index[i] + 1
}

Mres_PC <- Residuals_PC
Mres <- Mres_PC%*%t(spectra_PCA_loadings)

5 PCA on the effect matrices

5.1 PCA on the pure effect matrices

###################################################
## PCA on the effect matrices
###################################################

## Fixed effects -------------------------------
loadingsFix <- vector(mode = "list", length = length(Mfix))
for (i in 1:length(Mfix)){
  print(names(Mfix[i]))
  dimnames(Mfix[[i]]) <- dimnames(spectmat)
  res_pca <- SVDforPCA(Mfix[[i]])
  res_pca$scores <- round(res_pca$scores, 6)
  print(DrawScores(res_pca, size=3,
                         main = paste("Scores plot",
                                      names(Mfix[i])),
                   axes=c(1,2), drawNames=FALSE))

  barplot(res_pca$var[1:10], main  = "scree plot")

  plot(res_pca$loadings[,1], type="l", 
       main = paste("Loadings plot",names(Mfix[i])))
    
}

## [1] "Time"

## [1] "Tube"

# Time
#####################
res_pca <- SVDforPCA(Mfix$Time)

df <- data.frame(PC = as.character(1:7),var = res_pca$var[1:7])
screeplotTime <- ggplot(df, aes(y=0,yend=var,x=PC,
                          xend=PC))+ geom_segment()+
                   labs(title= "Scree plot",
                        x = "PC", y="% var")+theme_classic()
           
# screeplotTime

loadTime <- DrawLoadings(res_pca, type.obj = "PCA",createWindow = F,
          main = paste("Loadings plot"), 
          xaxis_type = "numeric",
                       axes = c(1), loadingstype = "s", num.stacked = 2,
                       xlab = "ppm", ylab = "", ang = "0",
                       nxaxis = 10)[[1]]


# Tube
#####################

res_pca <- SVDforPCA(Mfix$Tube)

df <- data.frame(PC = as.character(1:7),var = res_pca$var[1:7])
screeplotTube <- ggplot(df, aes(y=0,yend=var,x=PC,
                          xend=PC))+ geom_segment()+
                   labs(title= "Scree plot",
                        x = "PC", y="% var")+theme_classic()
           
# screeplotTube

loadTube <- DrawLoadings(res_pca, type.obj = "PCA",createWindow = F,
          main = paste("Loadings plot"), 
          xaxis_type = "numeric",
                       axes = c(1), loadingstype = "s", num.stacked = 2,
                       xlab = "ppm", ylab = "", ang = "0",
                       nxaxis = 10)[[1]]


## Random effects -------------------------------
loadingsRand <- vector(mode = "list", length = length(Mrand))
for (i in 1:length(Mrand)){
  print(names(Mrand[i]))
  dimnames(Mrand[[i]]) <- dimnames(spectmat)
  res_pca <- SVDforPCA(Mrand[[i]])
  res_pca$scores <- round(res_pca$scores, 6)
  print(DrawScores(res_pca, size=3, color = design[,names(Mrand[i])],
                         main = paste("Scores plot",names(Mrand[i])),
                   axes=c(1,2), drawNames=FALSE))

  print(DrawScores(res_pca, size=3, color = design[,names(Mrand[i])],
             main = paste("Scores plot",names(Mrand[i])), 
             axes=c(3,4), drawNames=FALSE))

  barplot(res_pca$var[1:10], main  = "scree plot")

  plot(res_pca$loadings[,1], type="l", 
       main = paste("Loadings plot",names(Mrand[i])))
  plot(res_pca$loadings[,2], type="l", 
       main = paste("Loadings plot",names(Mrand[i])))
 
  load <- DrawLoadings(res_pca, axes=c(1,2), type.obj = "PCA", 
             loadingstype = "s",  xlab = "ppm",
             xaxis_type = "character",
             main = paste("Loadings plot:",
                          names(Mrand[i]), 
                          "effet matrix"))
  
  load
  loadingsRand[[i]] <-  load
}

## [1] "Sampling"

## [1] "Volunteer"

# Sampling
#####################
res_pca <- SVDforPCA(Mrand$Sampling)

df <- data.frame(PC = as.character(1:7),var = res_pca$var[1:7])
screeplotSampling <- ggplot(df, aes(y=0,yend=var,x=PC,
                          xend=PC))+ geom_segment()+
                   labs(title= "Scree plot",
                        x = "PC", y="% var")+theme_classic()
           
# screeplotSampling

loadSampling <- DrawLoadings(res_pca, type.obj = "PCA",
                             createWindow = F,
          main = paste("Loadings plot"), 
          xaxis_type = "numeric",
          axes = c(1,2), loadingstype = "s", num.stacked = 3,
          xlab = "ppm", ang = "0",ylab = "",
          nxaxis = 10)

# Volunteer
#####################
res_pca <- SVDforPCA(Mrand$Volunteer)

df <- data.frame(PC = as.character(1:7),var = res_pca$var[1:7])
screeplotVolunteer <- ggplot(df, aes(y=0,yend=var,x=PC,
                          xend=PC))+ geom_segment()+
                   labs(title= "Scree plot",
                        x = "PC", y="% var")+theme_classic()
           
# screeplotVolunteer

loadVolunteer <- DrawLoadings(res_pca, type.obj = "PCA",
                              createWindow = F,
          main = paste("Loadings plot"), 
          xaxis_type = "numeric",
          axes = c(1,2), loadingstype = "s", 
          num.stacked = 3,
          xlab = "ppm", ang = "0",ylab = "",
          nxaxis = 10)


# Residuals
#####################
res_pca_Mres <- SVDforPCA(Mres)
res_pca_Mres$scores <- round(res_pca_Mres$scores, 6)

pander("scores and loadings of the residuals")

scores and loadings of the residuals

DrawLoadings(res_pca_Mres, axes = 1, xaxis_type = "character",main = "Loadings plot for the Residuals effect matrix")

## [[1]]

# pdf(file.path(out_path, "scores-Mres.pdf"))
DrawScores(res_pca_Mres, main = "Scores plot for the Residuals effect matrix")

# dev.off()

5.2 PCA on the augmented effect matrices

5.2.1 addSegments function

addSegments <- function(group, data, pch=16, main = NULL, 
                        col = rainbow(n = length(unique(group))), ...) {
  
  group <- as.factor(group)
  
  plot(data$x, data$y, col=col[group], pch=pch, main=main, ...)
  
  group <- as.factor(group)
  xcent <- tapply(data[,1], group, FUN=mean)
  ycent <- tapply(data[,2], group, FUN=mean)
  centers <- data.frame(xcent=xcent, ycent=ycent)

  mapply(FUN = points, x = centers$xcent, 
         y = centers$ycent, col = col, 
         MoreArgs = list(pch=20, cex=0.7))

  submatrices <- split(x=data, f=group)
  for (i in 1:nlevels(group)){
    mapply(FUN = segments, x1 = submatrices[[i]]$x, 
           y1 = submatrices[[i]]$y, col = col[i],
         MoreArgs = list(x0 = centers$xcent[i], 
                         y0 = centers$ycent[i]))
  }
  
}

5.2.2 ASCA-E

• Tube: Mu + E
• Time: Mt + E
• Volunteer: Mv + Ms
• Sampling: Ms + E

# correction factors ++++++
a <- nlevels(design$Tube)
b <- nlevels(design$Time)
c <- nlevels(design$Volunteer)
d <- nlevels(design$Sampling)

#################################
###  Augmented time effect
#################################

pander("Augmented Time effect")

Augmented Time effect

res_pca <- SVDforPCA(Mfix$Time)
df1 <- (b-1)
df2 <- (a*b*c*d-a-b-c*d+2)
Fstat <- qf(.95, df1=df1, df2=df2) 
coef <- sqrt(Fstat*df1/df2)

res_pca$scores[,1:2] <- (Mfix$Time + Mres*coef)%*%res_pca$loadings[,1:2]

col <- c("1"=violetred, "2" = darkblue)
pch <- c("1"= 2, "2"= 4)
index <- c(84, 85, 23, 87, 88)

df <- as.data.frame(res_pca$scores)

Time_scores <- ggplot(aes(y=PC1, x= 1:nrow(df), color=Time, shape = Tube),data=df)+
  geom_point() +
  ggtitle(label = "PCA Score on augmented Time effect matrix")+
  xlab("Observation number")+  ylim(low=-2.1, high=2.1) +
  geom_hline(yintercept=0, linetype=2, color="gray", size=0.7)+
  theme(plot.title = element_text(face="plain"),axis.text.x = element_text(size=10)) +
   annotate("text", y = res_pca$scores[index,1]+ 0.1*c(1,1,1,1,1), 
         x = index, label = rownames(res_pca$scores)[index])
Time_scores

#################################
###  Augmented tube effect
#################################

pander("Augmented Tube effect")

Augmented Tube effect

res_pca <- SVDforPCA(Mfix$Tube)
df1 <- (a-1)
df2 <- (a*b*c*d-a-b-c*d+2)
Fstat <- qf(.95, df1=df1, df2=df2) 
coef <- sqrt(Fstat*df1/df2)

res_pca$scores[,1:2] <- (Mfix$Tube + Mres*coef)%*%res_pca$loadings[,1:2]

col <- c("1"=violetred, "2" = darkblue)
pch <- c("1"= 2, "2"= 4)

index <- c(85, 80,79, 78,84,81,22,19,88,18, 87,21)

df <- as.data.frame(res_pca$scores)

Tube_scores <- ggplot(aes(y=PC1, x= 1:nrow(df), 
                          color=Tube, shape = Time),
                      data=df) +
  geom_point() +
  ggtitle(label = "PCA Score on augmented Tube effect matrix")+
  xlab("Observation number")+
  geom_hline(yintercept=0, linetype=2, color="gray", size=0.7)+
  theme(plot.title = element_text(face="plain"),
        axis.text.x = element_text(size=10))+
   annotate("text", 
            y = res_pca$scores[index,1]+ 0.2*
              c(1,1,1,-1,1,-1, 1,1,-1,-1,1,1), 
         x = index, label = rownames(res_pca$scores)[index])

Tube_scores

#################################
### Augmented Volunteer effect
#################################

pander("Augmented Volunteer effect")

Augmented Volunteer effect

res_pca <- SVDforPCA(Mrand$Volunteer)
df1 <- (c-1)
df2 <- (c*d-c)
Fstat <- qf(.95, df1=df1, df2=df2) 
coef <- sqrt(Fstat*df1/df2)

res_pca$scores[,1:2] <- (Mrand$Volunteer+Mrand$Sampling*coef)%*%
  res_pca$loadings[,1:2]


res_pca$scores <- round(res_pca$scores, 6)
Volunteer_scores <- DrawScores(res_pca, 
               main = "PCA Scores on augmented effect matrix", 
           color = Volunteer, pch=Volunteer, 
           drawNames = FALSE, drawPolygon  = TRUE, 
           noLegend = FALSE, size=2)+
        coord_cartesian(xlim = c(-50, 50), 
                        ylim = c(-17, 17))+  
  theme(legend.text=element_text(size=10),
        plot.title = element_text(face="plain"),
                    legend.key.height=unit(0.55,"line"))

Volunteer_scores

#################################
### Augmented Sampling effect
#################################
pander("Augmented Sampling effect")

Augmented Sampling effect

res_pca <- SVDforPCA(Mrand$Sampling)
df1 <- (c-1)
df2 <- (a*b*c*d-a-b-c*d+2)
Fstat <- qf(.95, df1=df1, df2=df2) 
coef <- sqrt(Fstat*df1/df2)

res_pca$scores[,1:2] <- (Mrand$Sampling +
                           Mres*coef)%*%res_pca$loadings[,1:2]

res_pca$scores <- round(res_pca$scores, 6)

pch <- c("1"= 1, "2"= 17, "3"=8)

col <- c(darkblue, "dodgerblue", turquoise, violetred, 
         limegreen,"orangered", "forestgreen", "gold", 
         "goldenrod4", "orange1", "maroon1",
         "red3")

VolSamp <- as.factor(paste0(Volunteer, Sampling))
Vol <- as.factor(substr(VolSamp,1,2))
pch <- c("1"= 2, "2"= 8)
names(col) <- levels(Volunteer)

xlab <- paste0("PC", 1, " (", round(res_pca$var[1], 2),"%)")
ylab <- paste0("PC", 2, " (", round(res_pca$var[2], 2), "%)")
    
df <- as.data.frame(res_pca$scores[,1:2])
Sampling_scores <- ggplot(df, aes(PC1, PC2, group = VolSamp, 
               shape = Sampling)) +   
  geom_point(size = 2, aes(colour = Volunteer)) +
  geom_line(aes(colour = Volunteer)) +
  scale_shape_manual(name = "Sampling", values=c(8, 0,17),
                           guide=guide_legend(order=1, shape = 1)) + 
  coord_cartesian(xlim = c(-26, 26), ylim = c(-12, 12))   + 
  ggplot2::labs(title = "PCA Scores on augmented effect matrix", 
                x = xlab, y = ylab) +
  ggplot2::geom_vline(xintercept = 0, size = 0.1) + 
  ggplot2::geom_hline(yintercept = 0, size = 0.1) + 
  ggplot2::theme_bw() +
      ggplot2::theme(panel.grid.major = 
                       ggplot2::element_line(color = "gray60",
                       size = 0.2), 
                     panel.grid.minor = ggplot2::element_blank(),
                     panel.background = 
                       ggplot2::element_rect(fill = "gray98")) +   theme(legend.text=element_text(size=7),
                    legend.key.height=unit(0.55,"line"))


Sampling_scores

5.2.3 Thesis chapter output

a <- grid.arrange(screeplotTime, Time_scores,
                 loadTime, nrow=1,widths=c(0.3, 1,  1),
             top=textGrob("Time effect matrix",
                          gp=gpar(fontsize=20,font=2)))

b <-  grid.arrange(screeplotTube,
                  Tube_scores,loadTube, 
                  nrow=1,widths=c(0.3, 1,  1),
             top=textGrob("Tube effect matrix",
                          gp=gpar(fontsize=20,font=2)))

d <- grid.arrange(screeplotVolunteer,
            Volunteer_scores, loadVolunteer[[1]],
              nrow=1,widths=c(0.3, 1,  1),
             top=textGrob("Volunteer effect matrix",
                          gp=gpar(fontsize=20,font=2)))

c <- grid.arrange(screeplotSampling,
               Sampling_scores,loadSampling[[1]],
              nrow=1,widths=c(0.3, 1,  1),
             top=textGrob("Sampling effect matrix",
                          gp=gpar(fontsize=20,font=2)))

# Thesis chapter output
# pdf(file.path(fig_path,paste0(Vers,
#            "HSD_Scores_Loadings_EffectMat.pdf")),
#          height = 20, width = 14)
# ggarrange(a,b,c,d, nrow=5, labels = c("A", "B", "C", "D"))
# dev.off()

5.2.4 Residuals

res_pca <- SVDforPCA(Mres)

df <- data.frame(PC = as.character(1:length(res_pca$var)),
                 var = res_pca$var)

df <- df[1:7,]

screeplot_resid <- ggplot(df, aes(y=0,yend=var,x=PC,
                          xend=PC))+ geom_segment()+
          labs(title= "Scree plot",
                        x = "PC", y="% var")+theme_classic()

index <- c(85,  84, 80, 81, 79, 78, c(18,19, 23,  22))

scores_resid <- DrawScores(res_pca, drawNames = FALSE, 
            color = Volunteer, 
           pch = Volunteer, main ="PCA scores", 
           size = 3) + 
  annotate("text", y = (res_pca$scores[index,2]+0.3 +
                                0.4*c( 1 , 1, 1, 1,-1,1,1,1,1,1)),
                         x = res_pca$scores[index,1]+0.3,
                         label = rownames(res_pca$scores[index,1:2]))+ 
   theme(legend.text=element_text(size=10),
         legend.key.height=unit(0.7,"line")) 


scores_resid <- scores_resid + coord_cartesian(xlim = c(-9, 9), 
                                               ylim = c(-9, 9))

loadings_resid <- DrawLoadings(res_pca, type.obj = "PCA",
                               createWindow = F,
          main = paste("PCA loadings"), 
          xaxis_type = "numeric",
           axes = c(1,2), loadingstype = "s", 
          num.stacked = 2,
           xlab = "ppm", ylab = "", ang = "0",
           nxaxis = 10)[[1]]



# pdf(file.path(fig_path,
# paste0(Vers,"HSD_Scores_Loadings_Residuals.pdf")),
# height = 4, width = 11)

grid.arrange(scores_resid, loadings_resid,nrow=1,
             widths=c(0.8, 1))

# dev.off()

5.2.5 Journal article output

e <- grid.arrange(screeplot_resid, 
              scores_resid, loadings_resid,
              nrow=1,widths  = c(0.3, 1,  1),
             top=textGrob("Residual effect matrix",
                          gp=gpar(fontsize=20,font=2)))

# journal article output
# pdf(file.path(fig_path,paste0(Vers,
#            "HSD_Scores_Loadings_EffectMat.pdf")),
#          height = 25, width = 14)
ggarrange(a,b,c,d,e , nrow=5, labels = c("A", "B", "C", "D", "E"))

# dev.off()

6 Pourcentage de variance expliquée

6.1 Method from Nakagawa and Schielzeth (2012)

####################################################
# Pourcentage de variance expliquée
####################################################

# Random effects -----------------------------------
sigma2_res = Res_std_error_PC^2 # Residual
varcor_random_full <- as.data.frame(varcor_random_full)
Var_Mrand <- rbind(varcor_random_full, 
                   sigma2_res=sigma2_res) # only random effects
Var_Mrand <- data.matrix(Var_Mrand)



# fixed effect -----------------------------------
Var_Mfix <- c()
for (i in 1:length(fixNames)){
  #  variance of parameters values (population)
  Var_Mfix <- rbind(Var_Mfix, 
                    (apply(Mfix_PC[[i]], 2, var) *(n - 1) / n))
}
rownames(Var_Mfix) <- fixNames

rownames(Var_Mrand) <- c(ranNames, "Residuals")

Var_Mrand <- rbind(Volunteer=Var_Mrand["Volunteer",],
                   Sampling=Var_Mrand["Sampling",], 
                   Residuals=Var_Mrand["Residuals",])


# All var comp (fixed and random) +++++
var_comp <- rbind(Var_Mfix, Var_Mrand)


# log of var comp +++++
log_var_comp <- t(log1p(var_comp)) # log(x+1)
log_var_comp <- cbind(id=1:dim(log_var_comp)[1],
                      log_var_comp)
log_var_comp <- as.data.frame(log_var_comp)
log_var_comp$id <- str_pad(1:dim(log_var_comp)[1], 2, pad = "0")
log_var_comp <- melt(log_var_comp, id=c("id"))
log_var_comp$value <- as.numeric(log_var_comp$value)

names(log_var_comp) <- c("PC", "Effect", "Variance")
  

sum_var_comp <- rowSums(var_comp)

# Percent var expl by each effect -----
var_comp_m1_abs <- sum_var_comp
var_comp_m1 <- var_comp_m1_abs*100/sum(var_comp_m1_abs)
names(var_comp_m1) <- names(var_comp_m1_abs)


# pdf(file.path(out_path,"variance_components.pdf"), 
#     height = 3, width = 3, pointsize = 12)

xlim <- c(0, ceiling(max(var_comp_m1)/10) * 10)
par(mar=c(0.5,2,3,0.5)) 
barplot(var_comp_m1, 
        main="Variance components \n percentage",
        xaxt="n",las=2, col=c(darkblue, turquoise,violetred ,
                              limegreen, gray67), border = NA,
        legend = names(var_comp_m1), 
        args.legend = list(x="topleft", inset=c(0,0),
                           box.lty=0,cex = 0.7,
                           y.intersp = 0.8))

# dev.off()

table_var <- rbind(var_comp_m1_abs, var_comp_m1)
rownames(table_var) <- c("Sum variance for all responses", 
                         "percentage of variation")
colnames(table_var) <- rownames(var_comp)
pander(table_var)

	Time	Tube	Volunteer	Sampling	Residuals
Sum variance for all responses	0.2982	0.3888	385.8	150.9	7.543
percentage of variation	0.05472	0.07134	70.8	27.69	1.384

6.2 Chapter graph

library("RColorBrewer")
gg_color_hue <- function(n) {
  hues = seq(15, 375, length = n + 1)
  hcl(h = hues, l = 65, c = 100)[1:n]
}

p <- ggplot(log_var_comp, aes(Effect, Variance, group = PC))+
  ggtitle("HSD - Log of variance components")+
  scale_color_manual(values = setNames(gg_color_hue(nPC),
                                       levels(log_var_comp$PC)))

p <- p + geom_point(aes(colour = PC))+
  geom_line(aes(colour = PC,linetype=PC),size=0.5)+ 
  ylab(label = "log(variance)") +
  theme(legend.text=element_text(size=10),
        legend.key.height=unit(0.7,"line"))+
   scale_linetype_manual(values = 1:nPC)

names(var_comp_m1)[5] <- "Residuals"

p

# Thesis chapter output

# ggexport(p, filename = file.path(fig_path,paste0(Vers,"HSD_variance_components.pdf")),
#          height = 5, width = 6.5, pointsize=14)

6.3 Article journal graph

# journal article output
tab <- data.frame(Effect= names(var_comp_m1), 
                  pcvar = round(var_comp_m1,2))

var_comp_m1.table <- ggtexttable(tab,
                    cols = c("Effect", "Global var (%)"),
                    rows = NULL, theme = ttheme("classic",
                                                base_size = 10))

p <- p + annotation_custom(ggplotGrob(var_comp_m1.table),
                              xmin = 3.1, ymin = 3,
                              xmax = 0)
p

# ggexport(p, filename = file.path(fig_path,paste0(Vers,"HSD_variance_components.pdf")),
#          height = 5, width = 6.5, pointsize=16)

6.4 Method from Rejane’s thesis

(Rousseau 2011, ISBA, UCL)

fix_var <- (1/2)*2*(fixef_PC[2,]^2)

var_comp <- rbind(fix_var, Var_Mrand)


sum_var_comp <- apply(var_comp, 1, sum)
pander( "sum of the variance components over the responses")

sum of the variance components over the responses

pander(sum_var_comp)

fix_var	Volunteer	Sampling	Residuals
0.2982	385.8	150.9	7.543

var_comp_m3 <- sum_var_comp
var_comp_m3 <- var_comp_m3*100/sum(var_comp_m3)
names(var_comp_m3) <- rownames(var_comp)

# var_comp_m3 <- var_comp_m3*100/sum(var_comp_m3)
# names(var_comp_m3) <- rownames(var_comp)
# barplot(var_comp_m3)
# pander(var_comp_m3)

7 Bootstrap

# set up of the bootstrap
set.seed(2017)
nsim = 2000 # number of simulations

# name of the output 
name <- paste0("bootstrap_HumanSerumData_VSTT_TubeFix.RData")

# Set up -------------------
# formulas without the effet to test
null_formulas <- list( 
  Volunteer = "~ Time + Tube + (1|Sampling) ",
  Sampling = "~ Time + Tube +  (1|Volunteer)  ",
  Tube = "~ Time + (1|Volunteer) + (1|Volunteer:Sampling)",
  Time = "~  Tube + (1|Volunteer) + (1|Volunteer:Sampling)")

null_effects <- names(null_formulas)

REML <- c(TRUE, TRUE, FALSE, FALSE) # for each null formula
names(REML) <- names(null_formulas)

7.1 Likelihood Ratio calcul

##################################################
# True log-likelihood Ratio statistics      #
##################################################


# full model: MM_full
######################
# REML ++++++
loglik_PC_full_REML <- sapply(MM_full$merMod_obj, logLik, REML=T)
mean_loglik_PC_full_REML <- mean(loglik_PC_full_REML)

# ML ++++++
loglik_PC_full_ML <- sapply(MM_full$merMod_obj, logLik, REML=F)
mean_loglik_PC_full_ML <- mean(loglik_PC_full_ML)


loglik_PC_full <- matrix(NA, ncol = nPC,
               nrow=length(null_formulas), byrow = TRUE)

for (i in 1:length(REML)){
  if (REML[i]==TRUE){
    loglik_PC_full[i,] <- loglik_PC_full_REML
  }else {loglik_PC_full[i,] <- loglik_PC_full_ML}
}


mean_loglik_PC_full <- rep(mean_loglik_PC_full_ML, 
                           length(null_formulas))

mean_loglik_PC_full[REML] <- mean_loglik_PC_full_REML



# Null models  
######################
res.parlmer_NULL <- vector("list", length = length(null_formulas))
names(res.parlmer_NULL) <- names(REML) <- names(null_formulas)


for (i in 1:length(null_formulas)) {
  # run parlmer
  res.parlmer_NULL[[i]] <- parlmer(design, 
                outcomes, null_formulas[[i]], REML=REML[i])
}

# Save the results ------------
MM_PC_null <- lapply(res.parlmer_NULL, function(x) x[["merMod_obj"]])

ranNames <- sapply(res.parlmer_NULL, function(x) x[["ranNames"]])
Fixvarnames <- lapply(res.parlmer_NULL, function(x) x[["fixNames"]])

varcor_random <- vector(mode = "list", length = length(null_formulas))
fixef_PC <- vector(mode = "list", length = length(null_formulas))
modmat_fixed <- vector(mode = "list", length = length(null_formulas))
M0 <- vector(mode = "list", length = length(null_formulas))


for (i in 1:length(null_formulas)){
  varcor_random[[i]] <- sapply(MM_PC_null[[i]], 
                               function(x)
                                 as.data.frame(VarCorr(x))$vcov)
  
  rownames(varcor_random[[i]]) <- c(names(VarCorr(MM_PC_null[[i]][[1]])),
                                    "Residual")

  fixef_PC[[i]] <- sapply(MM_PC_null[[i]], fixef)

  modmat_fixed[[i]] <- model.matrix(MM_PC_null[[i]][[1]], type = "fixed")

  # intercept
  XM0 <- modmat_fixed[[i]]
  XM0[,-1] <- 0
  M0_PC <- XM0%*%fixef_PC[[i]]
  M0[[i]] <- M0_PC%*%t(spectra_PCA_loadings)
}


# Effect matrices computation ------------
index <- vector("list", length=length(null_formulas))
fixNames_int <- lapply(Fixvarnames, function(x) c("(Intercept)", x))

colnam <- lapply(modmat_fixed,  colnames)

index <- vector("list", length=length(Fixvarnames))
for (i in 1:length(null_formulas)){
  id <- c()
  for (k in 1:length(fixNames_int[[i]])){
    id <- c(id, grep(fixNames_int[[i]][[k]], colnam[[i]]))
  }
  index[[i]] <- id
}


Mfix <- Mfix_PC <- vector("list", length=length(null_formulas))
names(Mfix) <- names(fixNames_int)

for (i in 1:length(null_formulas)){
  XMfix = modmat_fixed[[i]]
  XMfix[,-c(index[[i]])] = 0
  Mfix_PC[[i]] = XMfix %*% fixef_PC[[i]] # Matrix of the Group effect
  # backtransform the PC to original coefficients
  Mfix[[i]] <- Mfix_PC[[i]]%*%t(spectra_PCA_loadings)
}

fitted_values_PC <- Mfix

######################
# compute the LRT 
######################

# objects initialisation ------------
Res_std_error_PC_null <- vector("list", length=length(null_formulas))
loglik_PC_null <- vector("list", length=length(null_formulas))
mean_loglik_PC_null <- c()
meanlog <- c()
sumlog <- c()




### compute the LRT ----------------------------
for (i in 1:length(null_formulas)){
  Res_std_error_PC_null[[i]]  <- sapply(MM_PC_null[[i]], sigma)

  # meanlog and sumlog
  loglik_PC_null[[i]] <- sapply(MM_PC_null[[i]], logLik, REML=REML[i])
  mean_loglik_PC_null[i] <- mean(loglik_PC_null[[i]])

  meanlog[i] <- -2*(mean_loglik_PC_null[i] /mean_loglik_PC_full[i])
  sumlog[i] <- 2*(sum(loglik_PC_full[i,] - loglik_PC_null[[i]]))
}

names(meanlog) <- names(null_formulas)
names(sumlog) <- names(null_formulas)


# Graphs ----------------------------

col1="blue"
col2="red"
par(mar=c(4,3,2,6))
dif <- vector(mode = "list")
for (i in 1:length(null_formulas)){
  # graphs
  mat <- cbind(loglik_PC_null[[i]], loglik_PC_full[i,])
  rownames(mat) <- paste0("PC", 1:nPC)
  col <- c(col1,col2)
  par(xpd=TRUE)
  barplot(t(mat), beside=T, ylab="Log-likelihood",
          cex.names=0.8, las=2, col=col, 
          main = paste("log-likelihoods",names(null_formulas)[i]),
          xpd=TRUE)

  legend("topright", legend = c("loglik restricted","loglik full"),
         fill = col, bty = "n", inset=c(-0.1,0))
  
   dif[[i]] <- 2*(mat[,2] - mat[,1])
  
  
}

difmat <- do.call(cbind, dif)
colnames(difmat) <- names(null_formulas)
col=c(darkblue,turquoise,violetred , limegreen)
barplot(t(difmat), beside = TRUE, col=col,
        main = "Log-likelihood ratios")
legend("topright", legend = names(null_formulas), 
       title = "Removed effect", col = col, pch=15)

7.2 Booststrap tests

### Bootstrap ---------------------------------------
# bootstrapLT input arguments:
# MM_null = MM_PC_null$Sampling
# useREML=TRUE
# null_formula <- null_formulas[[2]]

 
bootstrapLT <- function(null_effect, useREML, MM_null, null_formula) {
  
  # simulate y from null models --------
  nPC <- length(MM_null)
  simulatedY <- c()

  for (i in 1:nPC){
    ysim       <- unlist(simulate(MM_null[[i]], re.form=NA))
    simulatedY <- cbind(simulatedY, ysim)
  }
  

  y <- simulatedY
  dimnames(y) <- dimnames(outcomes)

  # build restricted model -------
  # print("null")
  f_null <- parlmer(design, outcomes = y, null_formula, REML=useREML)

  
  # build full model -------
  # print("full")
  f_full <- parlmer(design, outcomes = y, form, REML=useREML)

  MM_f_null <- f_null$merMod_obj
  MM_f_full <- f_full$merMod_obj

  # LR -------
  loglikelihood_null <- sapply(MM_f_null, logLik, REML=useREML)
  mean_loglikelihood_null <- mean(loglikelihood_null)

  loglikelihood_full <- sapply(MM_f_full, logLik, REML=useREML)
  mean_loglikelihood_full <- mean(loglikelihood_full)

  sumlog <- 2*(sum(loglikelihood_full - loglikelihood_null))
  sumlog

}


sumlog_boot <- vector("list", length=length(null_formulas))

system.time(
for (i in 1:length(null_formulas)){
 
  null_effect <- null_effects[i]
  print(null_effect)
  sumlog_boot[[i]] = replicate(nsim, 
                           bootstrapLT(null_effect = null_effect, 
                           MM_null = MM_PC_null[[null_effect]],
                           useREML =REML[null_effect],
                           null_formula = null_formulas[[null_effect]]),
                           simplify = TRUE)

}
)

save(sumlog_boot, sumlog, file = file.path(out_path, name))

load(file=file.path(out_path, name))

7.3 Drawings

7.3.1 Histogram drawings

names(sumlog_boot) <- casefold(names(null_formulas), upper = FALSE)

######################
# Time
######################
# pdf(file = file.path(fig_path, "HSD_hist_Time.pdf"),
#     width = 7, height = 5)
par(mar=c(2.1, 2.1, 2.1, 7.5), xpd=TRUE, mfrow=c(1,1))

df <- nPC *   1 # df for Time and Tube

m=hist(sumlog_boot$time, freq=F, breaks=100,
       xlab="Global Likelihood Ratio Statistic",
       xlim=range(sumlog["Time"], sumlog_boot$time),
       ylim = c(0,0.08),
       col = gray67,border = gray67,
       main = " Fixed Time effect", cex.main = 2.2)
lines(density(sumlog_boot$time), col=darkblue,lwd=2)
lines(dchisq(seq(0,max(sumlog_boot$time)), df), col= "limegreen", lwd = 4)
points(sumlog["Time"], 0, col="red", pch=19, lwd=6)
legend("topright",
       legend = c(paste0("True GLRT: ", round(sumlog["Time"],2)),
                  "Kernel density", paste0("chi2 distrib. (df=", df,")")),
       col = c("red", darkblue, "limegreen"), lty=c(NA,1,1),pch=c(19,NA,NA),
       inset=c(-0.2,0),box.lty=0, cex = 1.4, y.intersp = 0.8, lwd=c(4))

# dev.off()


######################
# Tube
######################
# pdf(file = file.path(fig_path, "HSD_hist_Tube.pdf"),
#     width = 7, height = 5)
par(mar=c(2.1, 2.1, 2.1, 7.5), xpd=TRUE, mfrow=c(1,1))

df <- nPC *   1 # df for Time and Tube

m=hist(sumlog_boot$tube, freq=F, breaks=100,
       xlab="Global Likelihood Ratio Statistic",
       xlim=range(sumlog["Tube"], sumlog_boot$tube),
       ylim = c(0,0.08),
       col = gray67,border = gray67,
       main = " Fixed Tube effect", cex.main = 2.2)
lines(density(sumlog_boot$tube), col=darkblue,lwd=2)
lines(dchisq(seq(0,max(sumlog_boot$tube)), df), col= "limegreen", lwd = 4)
points(sumlog["Tube"], 0, col="red", pch=19, lwd=6)
legend("topright",
       legend = c(paste0("True GLRT: ", round(sumlog["Tube"],2)),
                  "Kernel density", paste0("chi2 distrib. (df=", df,")")),
       col = c("red", darkblue, "limegreen"), lty=c(NA,1,1),pch=c(19,NA,NA),
       inset=c(-0.2,0),box.lty=0, cex = 1.4, y.intersp = 0.8, lwd=c(4))

# dev.off()


######################
# Volunteer
######################
# pdf(file = file.path(fig_path, "HSD_hist_Volunteer.pdf"),
#     width = 7, height = 5)
par(mar=c(2.1, 2.1, 2.1, 7.5), xpd=TRUE)
df <- nPC
m=hist(sumlog_boot$volunteer, freq=F, breaks=100,
       xlab="Global Likelihood Ratio Statistic",
       xlim=range(sumlog["Volunteer"], sumlog_boot$volunteer),
       col = gray67,border = gray67,
       main = " Random Volunteer effect",
       cex.main = 2.2)
lines(density(sumlog_boot$volunteer), col=darkblue,lwd=2)
# y <- (1/2*dchisq(seq(0,30), df)+
#         (1/2*dchisq(seq(0,30), 0)))

# lines(0:30,y, col= "limegreen", lwd = 4)

points(sumlog["Volunteer"], 0, col="red", pch=19, lwd=6)
legend("topright",
       legend = c(paste0("True GLRT: ", round(sumlog["Volunteer"],2)),
                  "Kernel density"),
       col = c("red", darkblue), lty=c(NA,1),pch=c(19,NA),
       inset=c(-0.2,0),box.lty=0, cex = 1.4, y.intersp = 0.8, lwd=c(4))

# dev.off()

######################
# Sampling
######################
# pdf(file = file.path(fig_path, "HSD_hist_Sampling.pdf"),
#     width = 7, height = 5)
par(mar=c(2.1, 2.1, 2.1, 7.5), xpd=TRUE)
df <- nPC
m=hist(sumlog_boot$sampling, freq=F, breaks=100,
       xlab="Global Likelihood Ratio Statistic",
       xlim=range(sumlog["Sampling"], sumlog_boot$sampling),
       col = gray67,border = gray67,
       main = " Random Sampling effect",
       cex.main = 2.2)
lines(density(sumlog_boot$sampling), col=darkblue,lwd=2)
# y <- (1/2*dchisq(seq(0,30), df)+
#         (1/2*dchisq(seq(0,30), 0)))

# lines(0:30,y, col= "limegreen", lwd = 4)

points(sumlog["Sampling"], 0, col="red", pch=19, lwd=6)
legend("topright",
       legend = c(paste0("True GLRT: ", round(sumlog["Sampling"],2)),
                  "Kernel density"),
       col = c("red", darkblue), lty=c(NA,1),pch=c(19,NA),
       inset=c(-0.2,0),box.lty=0, cex = 1.4, y.intersp = 0.8, lwd=c(4))

# dev.off()

7.3.2 bootstrap p-values

pval <- c()
for (i in 1:length(null_formulas)){
 pval[i] <- (sum(sumlog[i]<sumlog_boot[[i]])+1)/(nsim+1)
}

names(pval) <- names(null_formulas)
pander("p-values")

p-values

pander(pval)

Volunteer	Sampling	Tube	Time
0.0004998	0.0004998	0.0004998	0.001499

7.4 p-value based on chi2 distribution for fixed effects

pander("true GLLR:")

true GLLR:

sumlog["Time"]

##     Time 
## 42.36541

df <- 1*nPC
curve(dchisq(x, df=df), col='red', main = "Chi-Square Density Graph",
          from=0,to=60)

pander("p-value Time")

p-value Time

pchisq(sumlog["Time"], df=df, lower.tail=FALSE)

##         Time 
## 0.0001974175

pander("p-value Tube")

p-value Tube

pchisq(sumlog["Tube"], df=df, lower.tail=FALSE)

##         Tube 
## 4.544409e-09

7.4.1 Article journal graph

difmat <- do.call(cbind, dif)
difmat <- data.frame(PC=substr(rownames(difmat),3,4), difmat)
colnames(difmat) <- c("PC", names(null_formulas))
difmat <- gather(difmat, key=Effect, value = value, Time, Tube, Volunteer, Sampling)
difmat$Effect <- factor(difmat$Effect, levels = c("Time", "Tube", "Volunteer", "Sampling"))

difmat$Effect

##  [1] Time      Time      Time      Time      Time      Time      Time     
##  [8] Time      Time      Time      Time      Time      Time      Time     
## [15] Time      Tube      Tube      Tube      Tube      Tube      Tube     
## [22] Tube      Tube      Tube      Tube      Tube      Tube      Tube     
## [29] Tube      Tube      Volunteer Volunteer Volunteer Volunteer Volunteer
## [36] Volunteer Volunteer Volunteer Volunteer Volunteer Volunteer Volunteer
## [43] Volunteer Volunteer Volunteer Sampling  Sampling  Sampling  Sampling 
## [50] Sampling  Sampling  Sampling  Sampling  Sampling  Sampling  Sampling 
## [57] Sampling  Sampling  Sampling  Sampling 
## Levels: Time Tube Volunteer Sampling

LLRplot <- ggplot(data=difmat, aes(x=PC, y=value, fill = Effect)) +
  geom_bar(width=0.5,stat="identity", position=position_dodge(width=0.5)) +
  theme_classic()+labs(title="(Restricted) Log-likelihood Ratios") +  
  ylab(label="(R)LLR") +
  guides(fill=guide_legend(title="Removed effect"))


tab <- round(pval[c(4,3, 1,2)], 4)
tab[2:4] <- paste("<", round(pval[1:3], 4))
tab <- data.frame(Effect = names(tab), `p-value` = tab,
                  chi2 = c("<5e-04","<5e-04","-", "-"))
pval_boot.table <- ggtexttable(tab,cols = c("Effect",
                                            "Boostrapped p-value", 
                                            "Chi2 test"),
                                 rows = NULL)


LLR_pval_plot <- ggarrange(LLRplot, pval_boot.table,
                ncol = 1, nrow = 2,
                heights = c(1,  0.3), common.legend=TRUE)

# ggexport(LLR_pval_plot, filename = file.path(fig_path,"HSD_LLR_pval_plot.pdf"),
#          height = 6, width = 5)

LLR_pval_plot

# journal article output

p <- LLRplot + annotation_custom(ggplotGrob(pval_boot.table),
                              xmin = 13, ymin = 400)
p

# ggexport(p, filename = file.path(fig_path,paste0(Vers,"HSD_GLLR.pdf")),
#          width=7, height=4.5, pointsize=5)

7.4.2 Chapter graph

difmat <- do.call(cbind, dif)
difmat <- data.frame(PC=substr(rownames(difmat),3,4), difmat)
colnames(difmat) <- c("PC", names(null_formulas))
difmat <- gather(difmat, key=Effect, value = value, Volunteer, Sampling,     Tube, Time)
difmat$Effect <- as.factor(difmat$Effect)
difmat$Effect <- factor(difmat$Effect,levels(difmat$Effect)[c(4,1,3,2)])

difmat2 <- cbind(Time=difmat[difmat$Effect=="Time", "value"],
                 Tube=difmat[difmat$Effect=="Tube", "value"],
                 Volunteer=difmat[difmat$Effect=="Volunteer", "value"],
                 Sampling=difmat[difmat$Effect=="Sampling", "value"]
                 )

rownames(difmat2) <- c(1:dim(difmat2)[1])

gg_color_hue <- function(n) {
  hues = seq(15, 375, length = n + 1)
  hcl(h = hues, l = 65, c = 100)[1:n]
}

# pdf(file.path(fig_path, paste0(Vers,"HSD_GLLR.pdf")),width=8, 
#     height=6, pointsize = 14)


par(mar=c(4,4,4,4))

# plotting settings -------------------------------------------------------
ylim <- range(mat)*c(1,1.5)
angle1 <- rep(c(45,45,135), length.out=4)
angle2 <- rep(c(45,135,135), length.out=4)
density1 <- seq(5,35,length.out=4)
density2 <- seq(5,35,length.out=4)
angle1 <- rev(angle1)
angle2 <- rev(angle2)
density1 <- rev(density1)
density2 <- rev(density2)

barplot(t(difmat2), beside=TRUE,col = gg_color_hue(4),ylab="(R)LLR", 
        xlab="PC", 
         main = "(Restricted) Log-likelihood Ratios",
        angle=angle1, density=density1)
barplot(t(difmat2), beside=TRUE, add=TRUE, col = gg_color_hue(4),
        ylab="(R)LLR", xlab="PC", 
         main = "(Restricted) Log-likelihood Ratios",
        angle=angle2, density=density2)
legend("topright", legend = colnames(difmat2), 
       title = "Removed effect:", col = gg_color_hue(4),
       fill=gg_color_hue(4),angle=angle1, density=density1)
par(bg="transparent")
legend("topright", legend = colnames(difmat2), 
       title = "Removed effect:", col = gg_color_hue(4),
       fill=gg_color_hue(4),angle=angle2, density=density2)

# dev.off()

# pdf(file.path(fig_path, paste0(Vers,"HSD_pval.pdf")))
pval_boot.table

# dev.off()